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The European Physical Journal B

Nonlinear critical slowing down of the one-component Ginzburg-Landau field

Richard Bausch, Erich Eisenriegler, H. K. Janssen

Published Jun 1, 1979• 20 citations
Publisher
DOI
10.1007/bf01320218

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We examine the critical-point singularity of the nonlinear relaxation of purely dissipative systems. Applying the molecular-field approximation to the time-dependent Ginzburg-Landau model, it is shown that the critical exponent of the nonlinear relaxation time (${\ensuremath{\Delta}}^{(\mathrm{nl})}$) can be different from that of the linear one (${\ensuremath{\Delta}}^{(l)}$) even in ergodic systems. From the assumptions that scaling applies and that only the near-equilibrium fluctuations significantly affect the relaxation time, a scaling law ${\ensuremath{\Delta}}^{(\mathrm{nl})}={\ensuremath{\Delta}}^{(l)}\ensuremath{-}\ensuremath{\beta}$ is derived ($\ensuremath{\beta}$ is the critical exponent of the order). On the base of this scaling law we reanalyze the available Monte Carlo calculations, high-temperature series, and the critical-slowing-down experiment on Ni3{\mathrm{Ni}}_{3}Ni3​Mn.
A number of examples have demonstrated the failure of the Landau-Ginzburg-Wilson (LGW) paradigm in describing the competing phases and phase transitions of two-dimensional quantum magnets. In this paper, we argue that such magnets possess field theoretic descriptions in terms of their slow fluctuating orders provided certain topological terms are included in the action. These topological terms may thus be viewed as what goes wrong within the conventional LGW thinking. The field theoretic descriptions we develop are possible alternates to the popular gauge theories of such non-LGW behavior. Examples that are studied include weakly coupled quasi-one-dimensional spin chains, deconfined critical points in fully two-dimensional magnets, and two-component massless QED3{\mathrm{QED}}_{3}QED3​. A prominent role is played by an anisotropic O(4) nonlinear sigma model in three space-time dimensions with a topological theta term. Some properties of this model are discussed. We speculate that similar sigma model descriptions might exist for fermionic algebraic spin liquid phases.
We present a field-theoretical method to obtain consistently the equations of motion for small amplitude fluctuations of the order parameter directly in real time for a homogeneous, neutral BCS superconductor. This method allows to study the nonequilibrium relaxation of the order parameter as an initial value problem. We obtain the Ward identities and the effective actions for small phase the amplitude fluctuations to one-loop order. Focusing on the long wavelength, low-frequency limit near the critical point, we obtain the time-dependent Ginzburg-Landau effective action to one-loop order, which is nonlocal as a consequence of Landau damping. The nonequilibrium relaxation of the phase and amplitude fluctuations is studied directly in real time. The long-wavelength phase fluctuation (Bogoliubov-Anderson-Goldstone mode) is overdamped by Landau damping and the relaxation time scale diverges at the critical point, revealing critical slowing down.